Multifractal Analysis of a Class of Additive Processes with Correlated Non-Stationary Increments
Julien Barral, INRIA Rocquencourt, France
Jacques Lèvy Vèhel, NRIA Rocquencourt, France
Abstract
We consider a family of stochastic processes built from infinite
sums of independent positive random functions on R +.
Each of these functions increases linearly between two consecutive
negative jumps, with the jump points following a Poisson point
process on R +. The motivation for studying these
processes stems from the fact that they constitute simplified
models for TCP traffic on the Internet. Such processes bear some
analogy with Lévy processes, but they are more complex in the
sense that their increments are neither stationary nor
independent. Nevertheless, we show that their multifractal
behavior is very much the same as that of certain Lévy
processes. More precisely, we compute the Hausdorff multifractal
spectrum of our processes, and find that it shares the shape of
the spectrum of a typical Lévy process. This result yields a
theoretical basis to the empirical discovery of the multifractal
nature of TCP traffic.
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